Existence of positive weak solutions for (p, q)-Laplacian nonlinear systems

Abstract

We mainly consider the existence of a positive weak solution of the following system

$$\left\{ \begin{array}[c]{cc}-{\Delta}_{p}u+\left\vert u\right\vert^{p-2}u=\lambda\left[ g(x)a(u)+c(x)f(v)\right] & ~\text{in}~{\Omega},\\-{\Delta}_{q}v+\left\vert v\right\vert^{q-2}v=\mu\left[ g(x)b(v)+c(x)h(u)\right] & ~\text{in}~ {\Omega},\\ u=v=0 & ~\text{on}~\partial{\Omega}, \end{array} \right. $$

where Δ _p u= div(|∇u|^p−2∇u),p,q >1 and λ,μ are positive parameters, and Ω⊂R ^N is a bounded domain with smooth boundary ∂Ω and g,c are nonnegative and continuous functions and f,h,a,b are C ¹ nondecreasing functions satisfying a(0),b(0)≥0. We have proved the existence of a positive weak solution for λ, μ large when

$$\lim\limits_{x\rightarrow\infty}\frac{f\left[ M\left( h\left( x\right) \right)^{\frac{1}{q-1}}\right]}{x^{p-1}}=0 $$

for every M > 0.